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A motivic interpretation of Whittaker periods for \(\textrm{GL}_n\)

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Abstract

Admitting the existence of conjectural motives attached to cohomological irreducible cuspidal automorphic representations of \(\textrm{GL}_n\), we write down Raghuram and Shahidi’s Whittaker periods in terms of Yoshida’s fundamental periods when the base field is a totally real number field or a CM field.

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Notes

  1. Here \(H^{\pm }_\textrm{dR}({\mathcal {M}}^{[\rho \tau ]})\) play the roles of \(\rho H^{\pm }_\textrm{dR}({\mathcal {M}})\) introduced in [26] and [28]; the \(\rho \)-twisted modules \(\rho H^{\pm }_\textrm{dR}({\mathcal {M}})\) seem to be defined only when the base field of \({\mathcal {M}}\) is a subfield of \(\textbf{C}\) stable under the complex conjugation \(\rho \).

  2. Note that \(E^\sigma \textsf{F}^{(\tau , \rho )}=E^\sigma \textsf{F}^\tau \) when \(\tau :\textsf{F}\hookrightarrow \textbf{C}\) is a real embedding.

  3. For a CM field F, we have \(F^{(\tau ,\rho )}=F^\tau \) since \(\rho \tau (F)\) coincides with \(\tau (F)\).

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Acknowledgements

The authors are sincerely grateful to Tadashi Miyazaki for the valuable discussions and for providing us the preprint version of [15]. They would also express their sincere gratitude to the anonymous referee for his or her careful reading and many valuable comments.

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Authors and Affiliations

  1. Department of Mathematics, College of Liberal Arts, Tsuda University, 2-1-1 Tsuda-machi, Kodaira City, Tokyo, 187-8577, Japan

    Takashi Hara

  2. Department of Mathematics, School of System Design and Technology, Tokyo Denki University, 5 Senju Asahi-cho, Adachi-ku, Tokyo, 120-8551, Japan

    Kenichi Namikawa

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  1. Takashi Hara
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Correspondence to Kenichi Namikawa.

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T. Hara was supported by Grant-in-Aid for Scientific Research (C) Grant Number JP22K03237. K. Namikawa was supported by Grant-in-Aid for Scientific Research (C) Grant Number JP21K03207.

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Hara, T., Namikawa, K. A motivic interpretation of Whittaker periods for \(\textrm{GL}_n\). manuscripta math. 174, 303–353 (2024). https://doi.org/10.1007/s00229-023-01507-1

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